Transition accounting for India in a multi-sector dynamic general equilibrium model

Abstract

Using a quantitative methodology designed specifically for emerging economies, we measure the components of India’s economic growth over the period 1960–2005. Our approach accounts for time-varying parameters, transitional dynamics and non-linear trends. We find that increased productivity in the service sector, facilitated by a structural shift toward services, is the principal driver of India’s economic growth. Our measures also suggest that the allocation of inputs across sectors has not improved over this period, and in the case of labor appears to have significantly worsened. We further find that fluctuations in output around its trend are due primarily to fluctuations in sector-specific total factor productivity, with fluctuations in labor market distortions and labor taxes also playing important roles. In the period 1960–1980, productivity fluctuations in the agricultural sector are the dominant source of cycles. Since then, productivity fluctuations in the manufacturing and service sectors have been more important.

Appendix: Background calculations

1.1 Input prices

Assuming interiority, the first-order conditions for intermediate goods producers are

$$\begin{aligned} (1-\mu )\frac{p_{mt}Y_{mt}}{L_{mt}}=w_{mt}, \end{aligned}$$

(24a)

$$\begin{aligned} (1-\sigma )\frac{p_{st}Y_{st}}{L_{st}}=w_{st}, \end{aligned}$$

(24b)

$$\begin{aligned} (1-\alpha )\frac{p_{at}Y_{at}}{L_{at}}=w_{at}, \end{aligned}$$

(24c)

$$\begin{aligned} \alpha \frac{p_{at}Y_{at}}{K_{at}}=\mu \frac{p_{mt}Y_{mt}}{K_{mt}}(1+\kappa _{mt})=\sigma \frac{p_{st}Y_{st}}{K_{st}}(1+\kappa _{st}). \end{aligned}$$

(24d)

Under perfect competition and constant returns, the average cost of the final good will equal its price, and it follows from Eqs. (3a) to (3c) that

$$\begin{aligned} p_{t} & = \frac{1}{Y_{t}}\left[ p_{at}Y_{at}+p_{mt}Y_{mt}+p_{st}Y_{st}\right] \nonumber \\ & = \frac{1}{Y_{t}}\left[ p_{t}\psi _{t}Y_{t}+p_{t}\eta _{t}Y_{t}+p_{t}\theta _{t}Y_{t}\right] =1. \end{aligned}$$

(25)

Combining these results with Eqs. (3a) to (3c) produces Eqs. (4a–5c).

1.2 Input aggregation

Combining Eqs. (1a–2), and inserting Eqs. (5a–5c) produces

$$\begin{aligned} K_{at} &= \frac{\psi _{t}\alpha }{\zeta _{t}(1+\kappa _{t})}K_{t}, \\ K_{mt} & = \frac{\eta _{t}\mu (1+\kappa _{mt})}{\zeta _{t}(1+\kappa _{t})}K_{t}, \nonumber \\ K_{st} &= \frac{\theta _{t}\sigma (1+\kappa _{st})}{\zeta _{t}(1+\kappa _{t})}K_{t}, \nonumber \\ \kappa _{t} & \equiv \frac{1}{\zeta _{t}}\left[ \eta _{t}\mu \kappa _{mt}+\theta _{t}\sigma \kappa _{st}\right] , \nonumber \\ \zeta _{t} & \equiv \alpha \psi _{t}+\mu \eta _{t}+\sigma \theta _{t}, \nonumber \end{aligned}$$

(26)

and we can rewrite Eq. (2) as:

$$\begin{aligned} Y_{t}=\widehat{A_{t}}\Omega _{t}K_{t}^{\zeta _{t}}L_{at}^{\psi _{t}(1-\alpha )}L_{mt}^{\eta \left( 1-\mu \right) }L_{st}^{\theta _{t}\left( 1-\sigma \right) }, \end{aligned}$$

(27)

where

$$\begin{aligned} \widehat{A_{t}} & \equiv A_{f_{t}}A_{at}^{\psi _{t}}A_{mt}^{\eta _{t}}A_{st}^{\theta _{t}}\zeta _{t}^{-\zeta _{t}}\left( \alpha \psi _{t}\right) ^{\alpha \psi _{t}}\left( \eta _{t}\mu \right) ^{\eta _{t}\mu }\left( \theta _{t}\sigma \right) ^{\theta _{t}\sigma }, \\ \Omega _{t} & \equiv \frac{\left( 1+\kappa _{mt}\right) ^{\eta _{t}\mu }\left( 1+\kappa _{st}\right) ^{\theta _{t}\sigma }}{\left( 1+\kappa _{t}\right) ^{\zeta _{t}}}. \end{aligned}$$

Combining Eqs. (4a) to (4c) with Eq. (7) produces

$$\begin{aligned} \psi _{t}(1-\alpha )\frac{Y_{t}}{L_{at}}=\eta _{t}(1-\mu )\frac{Y_{t}}{L_{mt}}(1+\ell _{mt})=\theta _{t}(1-\sigma )\frac{Y_{t}}{L_{st}}(1+\ell _{st}). \end{aligned}$$

Letting $\ell$ denote the average labor market “distortion”,

$$\begin{aligned} \ell _{t}=\frac{1}{1-\zeta _{t}}\left[ \eta _{t}(1-\mu )\ell _{mt}+\theta _{t}(1-\sigma )\ell _{st}\right] , \end{aligned}$$

we can rewrite the preceding equations as

$$\begin{aligned} L_{at} & = \frac{\psi _{t}(1-\alpha )}{(1-\zeta _{t})(1+\ell _{t})}L_{t}, \\ L_{mt} & = \frac{\eta _{t}\left( 1-\mu \right) (1+\ell _{mt})}{(1-\zeta _{t})(1+\ell _{t})}L_{t}, \nonumber \\ L_{st} & = \frac{\theta _{t}(1-\sigma )(1+\ell _{st})}{(1-\zeta _{t})(1+\ell _{t})}L_{t}. \nonumber \end{aligned}$$

(28)

Inserting these results into Eq. (27) and rearranging to find Eq. (8):

$$\begin{aligned} Y_{t}=A_{t}^{1-\zeta _{t}}K_{t}^{\zeta _{t}}L_{t}^{1-\zeta _{t}}, \end{aligned}$$

with the term A defined as in the main text.

Combining Eqs. (4a), (7) and (28), we can rewrite the labor-leisure allocation condition as

$$\begin{aligned} \chi \frac{C_{t}L_{t}^{\gamma }}{N_{t}^{1+\gamma }} & = \psi _{t}(1-\alpha )\frac{Y_{t}}{L_{at}} \\ & = (1-\zeta _{t})(1+\ell _{t})(1-\tau _{lt})\frac{Y_{t}}{L_{t}}. \end{aligned}$$

This is Eq. (9) in the main text.

Combining Eqs. (3a) and (26) produces

$$\begin{aligned} \zeta _{t}\left( 1+\kappa _{t}\right) \frac{Y_{t}}{K_{t}}=r_{t}. \end{aligned}$$

(29)

Combining Eqs. (6) and (29) produces Eq. (11) in the main text.

1.3 Normalized production

Let lower case variables denote upper case variables divided by population and trend productivity, with $c_{t}=C_{t}/\left( A^{*}_{t}N_{t}\right)$, and so on. The one exception is labor hours, where $l_{t}=L_{t}/N_{t}$. Inserting these definitions into Eqs. (8) and (9), we get equation (15) in the main text:

$$\begin{aligned} y_{t} & = k_{t}^{\zeta _{t} }(a_{t}l_{t})^{1-\zeta _{t} }, \\ & = \frac{ \chi c_{t}l_{t}^{1+\gamma } }{(1-\zeta _t)(1+\ell _{t})(1-\tau _{lt}) }, \\ & = \left( k_{t}^{\zeta _{t}}a_{t}^{1-\zeta _{t}}\right) ^{\phi _t} c_{t}^{1-\phi _{t}}\left( \frac{1}{\chi }( 1-\zeta _{t}) (1+\ell _{t})( 1-\tau _{lt}) \right) ^{\phi _{t}-1}, \end{aligned}$$

with $a_{t}$ and $\phi _{t}$ defined as in the main text.

1.4 Linearization

Let hats (“$\widehat{\; \;}$”) denote deviations around the transition path. The tax rates, the allocation wedges (the $\kappa$’s and the $\ell$’s), the production parameters ($\zeta$ and $\phi$), and the depreciation rate ($\delta$) are expressed as level deviations; in these cases $\widehat{z}_{t}=z_{t}-z_{t}^{*}$. All other variables are expressed as log deviations; in these cases, $\widehat{z}_{t}=\ln \left( z_{t}/z_{t}^{*}\right)$.

Consider the Euler equation

$$\begin{aligned} \frac{1}{c_{t}}=\beta \frac{N_{t+1}}{N_{t}}E_{t}\left( \frac{1}{c_{t+1}}(X_{t+1}^{*})^{-1}\left[ 1+\zeta _{t+1}(1+\kappa _{t+1})\frac{y_{t+1}}{k_{t+1}}-\delta _{t+1}\right] \right) \frac{1}{1+\tau _{kt}}, \end{aligned}$$

We can rewrite this expression as

$$\begin{aligned} \frac{1}{c_{t}^{*}}\exp \left( -\widehat{c}_{t}\right) & =\frac{1}{1+\tau _{kt}^{*}+\widehat{\tau }_{kt}}\times \beta E_{t}\left( \frac{1}{c_{t+1}^{*}}\exp \left( -\widehat{c}_{t+1}\right) \left( X_{A,t+1}^{*}\right) ^{-1} \right. \\& \quad\times \left. \left[ 1+(\zeta _{t+1}^{*} +\widehat{\zeta }_{t+1})(1+\kappa _{t+1}^{*}+\widehat{\kappa }_{t+1})\frac{y_{t+1}^{*}}{k_{t+1}^{*}}\exp \left( \widehat{y}_{t+1}-\widehat{k}_{t+1}\right) -\left( \delta _{t+1}^{*}+\widehat{\delta }_{t+1}\right) \right] \right) , \end{aligned}$$

where $X_{A,t+1}^{*}=A_{t+1}^{*}/A_{t}^{*}$. Logging both sides, and assuming the deviations are small, one gets

$$\begin{aligned} -&\widehat{c}_{t}\approx \ln \left( \lambda _{2,t+1}\right) -\ln \left( 1+\tau _{kt}^{*}+\widehat{\tau }_{kt}\right) -E_{t}\Big \{ \widehat{c}_{t+1}\Big \} \\&+E_{t}\left\{ \ln \left( 1+(\zeta _{t+1}^{*}+\widehat{\zeta }_{t+1})(1+\kappa _{t+1}^{*}+\widehat{\kappa }_{t+1})\lambda _{1,t+1}\exp \left( \widehat{y}_{t+1}-\widehat{k}_{t+1}\right) -\left( \delta _{t+1}^{*}+\widehat{\delta }_{t+1}\right) \right) \right\} , \\ \lambda _{1t}&\equiv \frac{y_{t}^{*}}{k_{t}^{*}};\quad \lambda _{2t}\equiv \beta \frac{c_{t-1}^{*}}{c_{t}^{*}}\left( X_{A,t}^{*}\right) ^{-1}. \end{aligned}$$

Implicitly differentiating around trend values (“stars”, with “hats” set equal to zero), and noting that $1/\left[ 1+\zeta _{t+1}(1+\kappa _{t+1}^{*})\lambda _{1,t+1}-\delta _{t+1}\right] =\lambda _{2,t+1}/(1+\tau _{kt}^{*})$, we get

$$\begin{aligned} \widehat{c}_{t} & \approx E_{t}\left( \widehat{c}_{t+1}-\lambda _{3,t+1}\left( \frac{1}{\zeta _{t+1}^{*}}\widehat{\zeta }_{t+1}+\widehat{y}_{t+1}-\widehat{k}_{t+1}+\frac{1}{1+\kappa _{t+1}^{*}}\widehat{\kappa }_{t+1}\right) -\frac{\lambda _{2,t+1}}{1+\tau _{kt}^{*}}\widehat{\delta }_{t+1}\right) +\frac{1}{1+\tau _{kt}^{*}}\widehat{\tau }_{kt}, \nonumber \\ \lambda _{3t} & \equiv \frac{\zeta _{t+1}^{*}\lambda _{1,t+1}\lambda _{2,t+1}(1+\kappa _{t+1}^{*})}{1+\tau _{kt}^{*}}. \nonumber \end{aligned}$$

Following Eq. (23), we replace $\widehat{\tau }_{kt}$ with $\rho _{k}\widehat{\widetilde{\tau }}_{k,t-1}$.

Next, consider the capital accumulation equation

$$\begin{aligned} X_{t+1}^{*}k_{t+1}=\left( 1-\delta _{t}\right) k_{t}+y_{t}-c_{t}-g_{t}, \end{aligned}$$

which can be rewritten as

$$\begin{aligned} X_{t+1}^{*}k_{t+1}^{*}\exp \left( \widehat{k}_{t+1}\right) =\left( 1-\left( \delta _{t}^{*}+\widehat{\delta }_{t}\right) \right) k_{t}^{*}\exp \left( \widehat{k}_{t}\right) +y_{t}^{*}\exp \left( \widehat{y}_{t}\right) -c_{t}^{*}\exp \left( \widehat{c}_{t}\right) -g_{t}^{*}\exp \left( \widehat{g}_{t}\right) . \end{aligned}$$

Implicit differentiation yields

$$\begin{aligned} X_{t+1}^{*}k_{t+1}^{*}\widehat{k}_{t+1}=k_{t}^{*}\left( 1-\delta _{t}^{*}\right) \widehat{k}_{t}-k_{t}^{*}\widehat{\delta }_{t}+y_{t}^{*}\widehat{y}_{t}-c_{t}^{*}\widehat{c}_{t}-g_{t}^{*}\widehat{g}_{t}, \end{aligned}$$

or

$$\begin{aligned} \lambda _{4,t+1}\widehat{k}_{t+1} & =\left( 1-\delta _{t}^{*}\right) \widehat{k}_{t}+\lambda _{1t}\widehat{y}_{t}-\lambda _{5t}\widehat{c}_{t}-\lambda _{6t}\widehat{g}_{t}-\widehat{\delta }_{t}, \\ \lambda _{4t} & \equiv \frac{k_{t}^{*}}{k_{t+1}^{*}}X_{t}^{*};\quad \lambda _{5t}\equiv \frac{c_{t}^{*}}{k_{t}^{*}};\quad \lambda _{6t}\equiv \frac{g_{t}^{*}}{k_{t}^{*}}. \end{aligned}$$

To fill out these two difference equations, we substitute for output, using

$$\begin{aligned} y_{t}=\left( \frac{1}{\chi }\left( 1-\zeta _{t}\right) \left( 1-\tau _{lt}\right) (1+\ell _{t})\right) ^{\phi _{t}-1}\left( k_{t}^{\zeta _{t}}a_{t}^{1-\zeta _{t}}\right) ^{\phi _{t}}c_{t}^{1-\phi _{t}}. \end{aligned}$$

(30)

Linearizing this equation requires us to consider the effects of the exponent deviations $\widehat{\phi }_{t}$ and $\widehat{\zeta }_{t}$. To see how this works, consider another expression for output:

$$\begin{aligned} y_{t}=k_{t}^{\zeta _{t}}(a_{t}l_{t})^{1-\zeta _{t}}. \end{aligned}$$

This equality can be rewritten as

$$\begin{aligned} \frac{y_{t}}{y_{t}^{*}}=\frac{k_{t}^{\zeta _{t}^{*}+\widehat{\zeta }_{t}}(a_{t}l_{t})^{1-(\zeta _{t}^{*}+\widehat{\zeta }_{t})}}{\left( k_{t}^{*}\right) ^{\zeta _{t}^{*}}(a_{t}^{*}l_{t}^{*})^{1-\zeta _{t}^{*}}}=\left( \frac{k_{t}}{k_{t}^{*}}\right) ^{\zeta _{t}^{*}}\left( \frac{a_{t}l_{t}}{a_{t}^{*}l_{t}^{*}}\right) ^{1-\zeta _{t}^{*}}k_{t}^{\widehat{\zeta }_{t}}(a_{t}l_{t})^{-\widehat{\zeta }_{t}}, \end{aligned}$$

and taking logs yields

$$\begin{aligned} \widehat{y}_{t} & = \zeta _{t}^{*}\widehat{k}_{t}+(1-\zeta _{t}^{*})(\widehat{a}_{t}+\widehat{l}_{t})+\widehat{\zeta }_{t}\ln \left( k_{t}\right) -\widehat{\zeta }_{t}\left( \ln (l_{t})+\ln (a_{t})\right) \\ & \approx \zeta _{t}^{*}\widehat{k}_{t}+(1-\zeta _{t}^{*})(\widehat{a}_{t}+\widehat{l}_{t})+\left[ \ln \left( k_{t}^{*}\right) -\ln (l_{t}^{*})\right] \widehat{\zeta }_{t}, \end{aligned}$$

as $\ln (a_{t}^{*})=0$.

To apply this approach to Eq. (30), we log both sides and implicitly differentiate:

$$\begin{aligned} \widehat{y}_{t} & = \phi _{t}^{*}\zeta _{t}^{*}\widehat{k}_{t}+\phi _{t}^{*}(1-\zeta _{t}^{*})\widehat{a}_{t}+(1-\phi _{t}^{*})\left( \widehat{c}_{t}+\frac{1}{1-\zeta _{t}^{*}}\widehat{\zeta }_{t}+\frac{1}{1-\tau _{lt}^{*}}\widehat{\tau }_{lt}-\frac{1}{1+\ell _{t}^{*}}\widehat{\ell }_{t}\right) \\& \quad +\phi _{t}^{*}\ln \left( \frac{k_{t}^{*}}{a_{t}^{*}}\right) \widehat{\zeta }_{t}+\left[ \zeta _{t}^{*}\ln (k_{t}^{*})+(1-\zeta _{t}^{*})\ln (a_{t}^{*})+\ln \left( \frac{1}{\chi c_{t}^{*}}\left( 1-\zeta _{t}^{*}\right) \left( 1-\tau _{lt}^{*}\right) (1+\ell _{t}^{*})\right) \right] \widehat{\phi }_{t} \\ & = \phi _{t}^{*}\zeta _{t}^{*}\widehat{k}_{t}+\phi _{t}^{*}(1-\zeta _{t}^{*})\widehat{a}_{t}+(1-\phi _{t}^{*})\left( \widehat{c}_{t}+\frac{1}{1-\tau _{lt}^{*}}\widehat{\tau }_{lt}-\frac{1}{1+\ell _{t}^{*}}\widehat{\ell }_{t}\right) \\& \quad +\left[ \frac{1-\phi _{t}^{*}}{1-\zeta _{t}^{*}}+\phi _{t}^{*}\ln \left( k_{t}^{*}\right) \right] \widehat{\zeta }_{t}+\left[ \zeta _{t}^{*}\ln (k_{t}^{*})+\ln \left( \frac{1}{\chi c_{t}^{*}}\left( 1-\zeta _{t}^{*}\right) \left( 1-\tau _{lt}^{*}\right) (1+\ell _{t}^{*})\right) \right] \widehat{\phi }_{t}. \end{aligned}$$

Next, we substitute for the components of $\widehat{a}_{t}$, $\widehat{\kappa }_{t}, \widehat{\ell }_{t}, \widehat{\zeta }_{t}$ and $\widehat{\phi }_{t}$. Consider first the expression for total factor productivity, $\widehat{a}_{t}$. It follows from Eq. (10) that

$$\begin{aligned} \frac{A_{t}^{1-(\zeta _{t}^{*}+\widehat{\zeta }_{t})}}{\left( A_{t}^{*}\right) ^{1-\zeta _{t}^{*}}}\equiv \frac{A_{ft}A_{at}^{\psi _{t}^{*}+\widehat{\psi }_{t}}A_{mt}^{\eta _{t}^{*}+\widehat{\eta }_{t}}A_{st}^{\theta _{t}^{*}+\widehat{\theta }_{t}}\Omega _{t}\Upsilon _{t}\Delta _{t}}{A_{ft}^{*}\left( A_{at}^{*}\right) ^{\psi _{t}^{*}}\left( A_{mt}^{*}\right) ^{\eta _{t}^{*}}\left( A_{st}^{*}\right) ^{\theta _{t}^{*}}\Omega _{t}^{*}\Upsilon _{t}^{*}\Delta _{t}^{*}}. \end{aligned}$$

Taking logs yields

$$\begin{aligned} \left( 1-\zeta _{t}^{*}\right) \widehat{a}_{t}-\widehat{\zeta }_{t}\ln \left( A_{t}\right) & = \psi _{t}^{*}\widehat{a}_{at}+\eta _{t}^{*}\widehat{a}_{mt}+\theta _{t}^{*}\widehat{a}_{st}+\widehat{\psi }_{t}\ln \left( A_{at}\right) +\widehat{\eta }_{t}\ln \left( A_{mt}\right) +\widehat{\theta }_{t}\ln \left( A_{st}\right) \\& \quad +\, \widehat{a}_{ft}+\widehat{\Omega }_{t}+\widehat{\Upsilon }_{t}+\widehat{\Delta }_{t}, \end{aligned}$$

or

$$\begin{aligned} \widehat{a}_{t} & \approx \frac{1}{1-\zeta _{t}^{*}}\left( \psi _{t}^{*}\widehat{a}_{at}+\eta _{t}^{*}\widehat{a}_{mt}+\theta _{t}^{*}\widehat{a}_{st}+\ln \left( A_{at}^{*}\right) \widehat{\psi }_{t}+\ln \left( A_{mt}^{*}\right) \widehat{\eta }_{t}+\ln \left( A_{st}^{*}\right) \widehat{\theta }_{t}+\ln \left( A_{t}^{*}\right) \widehat{\zeta }_{t}\right. \\& \quad \left. +\, \widehat{a}_{ft}+\widehat{\Omega }_{t}+\widehat{\Upsilon }_{t}+\widehat{\Delta }_{t}\right) . \end{aligned}$$

Continuing, it follows from the definition of $\Omega _{t}$ that:

$$\begin{aligned} \frac{\Omega _{t}}{\Omega _{t}^{*}}\equiv \frac{\left( 1+\kappa _{mt}^{*}+\widehat{\kappa }_{mt}\right) ^{\left( \eta _{t}^{*}+\widehat{\eta }_{t}\right) \mu }\left( 1+\kappa _{st}^{*}+\widehat{\kappa }_{st}\right) ^{\left( \theta _{t}^{*}+\widehat{\theta }_{t}\right) \sigma }}{\left( 1+\kappa _{t}^{*}+\widehat{\kappa }_{t}\right) ^{\zeta _{t}^{*}+\widehat{\zeta }_{t}}}\cdot \frac{\left( 1+\kappa _{t}^{*}\right) ^{\zeta _{t}^{*}}}{\left( 1+\kappa _{mt}^{*}\right) ^{\eta _{t}^{*}\mu }\left( 1+\kappa _{st}^{*}\right) ^{\theta _{t}^{*}\sigma }}. \end{aligned}$$

Taking logs, and then implicitly differentiating, yields

$$\begin{aligned} \widehat{\Omega }_{t} & = \eta _{t}^{*}\mu \ln \left( \frac{1+\kappa _{mt}^{*}+\widehat{\kappa }_{mt}}{1+\kappa _{mt}^{*}}\right) +\theta _{t}^{*}\sigma \ln \left( \frac{1+\kappa _{st}^{*}+\widehat{\kappa }_{st}}{1+\kappa _{st}^{*}}\right) -\zeta _{t}^{*}\ln \left( \frac{1+\kappa _{t}^{*}+\widehat{\kappa }_{t}}{1+\kappa _{t}^{*}}\right) \\& \quad +\widehat{\eta }_{t}\mu \ln \left( 1+\kappa _{mt}^{*}+\widehat{\kappa }_{mt}\right) +\widehat{\theta }_{t}\sigma \ln \left( 1+\kappa _{st}^{*}+\widehat{\kappa }_{st}\right) -\widehat{\zeta }_{t}\ln \left( 1+\kappa _{t}^{*}+\widehat{\kappa }_{t}\right) \\ & \approx \frac{\eta _{t}^{*}\mu }{1+\kappa _{mt}^{*}}\widehat{\kappa }_{mt}+\frac{\theta _{t}^{*}\sigma }{1+\kappa _{st}^{*}}\widehat{\kappa }_{st}-\frac{\zeta _{t}^{*}}{1+\kappa _{t}^{*}}\widehat{\kappa }_{t} \\& \quad +\mu \ln \left( 1+\kappa _{mt}^{*}\right) \widehat{\eta }_{t}+\sigma \ln \left( 1+\kappa _{st}^{*}\right) \widehat{\theta }_{t}-\ln \left( 1+\kappa _{t}^{*}\right) \widehat{\zeta }_{t}. \end{aligned}$$

Similarly,

$$\begin{aligned} \widehat{\Upsilon }_{t} & \approx \frac{\eta _{t}^{*}(1-\mu )}{1+\ell _{mt}^{*}}\widehat{\ell }_{mt}+\frac{\theta _{t}^{*}(1-\sigma )}{1+\ell _{st}^{*}}\widehat{\ell }_{st}-\frac{1-\zeta _{t}^{*}}{1+\ell _{t}^{*}}\widehat{\ell }_{t} \\& \quad +(1-\mu )\ln \left( 1+\ell _{mt}^{*}\right) \widehat{\eta }_{t}+(1-\sigma )\ln \left( 1+\ell _{st}^{*}\right) \widehat{\theta }_{t}+\ln \left( 1+\ell _{t}^{*}\right) \widehat{\zeta }_{t}. \end{aligned}$$

Finally, it follows from Eqs. (10) and (18) that

$$\begin{aligned} \Delta _{t}A_{ft}\equiv \left( \alpha ^{\alpha }\left( 1-\alpha \right) ^{1-\alpha }\right) ^{\psi _{t}}\left( \mu ^{\mu }\left( 1-\mu \right) ^{1-\mu }\right) ^{\eta _{t}}\left( \sigma ^{\sigma }\left( 1-\sigma \right) ^{1-\sigma }\right) ^{\theta _{t}}\zeta _{t}^{-\zeta _{t}}(1-\zeta _{t})^{\zeta _{t}-1}, \end{aligned}$$

so that

$$\begin{aligned} \, \widehat{a}_{ft}+\widehat{\Delta }_{t} & \approx \ln \left( \alpha ^{\alpha }\left( 1-\alpha \right) ^{1-\alpha }\right) \widehat{\psi }_{t}+\ln \left( \mu ^{\mu }\left( 1-\mu \right) ^{1-\mu }\right) \widehat{\eta }_{t}+\ln \left( \sigma ^{\sigma }\left( 1-\sigma \right) ^{1-\sigma }\right) \widehat{\eta }_{t} \\& \quad -\zeta _{t}^{*}\frac{1}{\zeta _{t}^{*}}\widehat{\zeta }_{t}-\widehat{\zeta }_{t}\ln \left( \zeta _{t}^{*}\right) +(\zeta _{t}^{*}-1)\frac{1}{1-\zeta _{t}^{*}}\left( -\widehat{\zeta }_{t}\right) +\widehat{\zeta }_{t}\ln \left( 1-\zeta _{t}^{*}\right) \\ & = \ln \left( \alpha ^{\alpha }\left( 1-\alpha \right) ^{1-\alpha }\right) \widehat{\psi }_{t}+\ln \left( \mu ^{\mu }\left( 1-\mu \right) ^{1-\mu }\right) \widehat{\eta }_{t}+\ln \left( \sigma ^{\sigma }\left( 1-\sigma \right) ^{1-\sigma }\right) \widehat{\eta }_{t} \\& \quad +\ln \left( \frac{1-\zeta _{t}^{*}}{\zeta _{t}^{*}}\right) \widehat{\zeta }_{t}. \end{aligned}$$

Next, we consider the sectoral distortions, $\kappa _{t}$ and $\ell _{t}$. Recall the capital distortion:

$$\begin{aligned} \kappa _{t}=\frac{1}{\zeta _{t}}\left[ \eta _{t}\mu \kappa _{mt}+\theta _{t}\sigma \kappa _{st}\right] . \end{aligned}$$

Implicit differentiation yields

$$\begin{aligned} \widehat{\kappa }_{t} & \approx \frac{1}{\zeta _{t}^{*}}\left[ \mu \kappa _{mt}^{*}\widehat{\eta }_{t}+\eta _{t}^{*}\mu \widehat{\kappa }_{mt}+\sigma \kappa _{st}^{*}\widehat{\theta }_{t}+\theta _{t}^{*}\sigma \widehat{\kappa }_{st}\right] -\frac{1}{\left( \zeta _{t}^{*}\right) ^{2}}\left[ \eta _{t}^{*}\mu \kappa _{mt}^{*}+\theta _{t}^{*}\sigma \kappa _{st}^{*}\right] \widehat{\zeta }_{t} \\ & = \frac{1}{\zeta _{t}^{*}}\left[ \mu \kappa _{mt}^{*}\widehat{\eta }_{t}+\eta _{t}^{*}\mu \widehat{\kappa }_{mt}+\sigma \kappa _{st}^{*}\widehat{\theta }_{t}+\theta _{t}^{*}\sigma \widehat{\kappa }_{st}-\kappa _{t}^{*}\widehat{\zeta }_{t}\right] . \end{aligned}$$

Similarly

$$\begin{aligned} \widehat{\ell }_{t}\approx \frac{1}{1-\zeta _{t}^{*}}\left[ (1-\mu )\ell _{mt}^{*}\widehat{\eta }_{t}+\eta _{t}^{*}(1-\mu )\widehat{\ell }_{mt}+(1-\sigma )\ell _{st}^{*}\widehat{\theta }_{t}+\theta _{t}^{*}(1-\sigma )\widehat{\ell }_{st}+\ell _{t}^{*}\widehat{\zeta }_{t}\right] . \end{aligned}$$

Finally, we express the composite parameters $\zeta _{t}$ and $\phi _{t}$ as functions of the share parameters $\psi _{t}$, $\eta _{t}$ and $\theta _{t}$. Recall that

$$\begin{aligned} \zeta _{t} & \equiv \alpha \psi _{t}+\mu \eta _{t}+\sigma \theta _{t}, \\ \phi _{t} & \equiv \frac{1+\gamma }{\gamma +\zeta _{t}}, \\ 1 & = \psi _{t}+\eta _{t}+\theta _{t}, \end{aligned}$$

yielding

$$\begin{aligned} \widehat{\zeta }_{t} & \equiv \alpha \widehat{\psi }_{t}+\mu \widehat{\eta }_{t}+\sigma \widehat{\theta }_{t}, \\ \widehat{\phi }_{t} & \equiv -\frac{1+\gamma }{\left( \gamma +\zeta _{t}^{*}\right) ^{2}}\widehat{\zeta }_{t}, \\ \widehat{\eta }_{t} & = -\widehat{\psi }_{t}-\widehat{\theta }_{t}. \end{aligned}$$

1.5 Solving time-varying linear expectational difference equations

Our approach most closely follows that of Klein (2000) (and to a lesser extent Sims 2002), although we use the eigenvalue-eigenvector decomposition introduced by Blanchard and Kahn (1980) (as well as their notation), rather than the generalized Schur decomposition. We also incorporate elements of the MDS approaches used by Broze et al. (1985), Farmer (1993) and Farmer and Guo (1994), as well as insights from King et al. (2002).

1.5.1 The basic solution

Consider the following system:

$$\begin{aligned} E_{t}\left( {\mathbf {v}}_{t+1}\right) ={\mathbf {A}}_{t}{\mathbf {v}}_{t},\quad t=0,1,2..., \end{aligned}$$

(31)

where ${\mathbf {v}}_{t}$ is an $(n\times 1)$ vector, ${\mathbf {A}}_{t}$ is an $(n\times n)$ time-dependent matrix, and $E_{t}\left( {\mathbf {.}}\right)$ is the usual conditional expectations operator. The vector ${\mathbf {v}}_{t}$ can be decomposed into the $(n_{1}\times 1)$ control vector ${\mathbf {x}}_{t}$ and the $(n_{2}\times 1)$ state vector ${\mathbf {p}}_{t}$. In particular, ${\mathbf {p}}_{t}$ is restricted by

$$\begin{aligned}&{\mathbf {d}}_{t+1}\equiv {\mathbf {p}}_{t+1}-E_{t}({\mathbf {p}}_{t+1})\,\text{given},\quad t=0,1,2\ldots,\\&{\mathbf {p}}_{0}\,\text {given},\nonumber\end{aligned}$$

(32)

where $\left\{ {\mathbf {d}}_{t+1}\right\} _{t=0}^{\infty }$ is a covariance stationary Martingale difference sequence. In contrast, ${\mathbf {x}}_{t}$ needs only to obey some standard boundedness conditions; ${\mathbf {g}}_{t+1}\equiv {\mathbf {x}}_{t+1}-E_{t}\left( {\mathbf {x}}_{t+1}\right)$ is otherwise unrestricted. Using these definitions, we can rewrite Eq. (31) as

$$\begin{aligned} \left( \begin{array}{c} {\mathbf {x}}_{t+1} \\ {\mathbf {p}}_{t+1}\end{array}\right) ={\mathbf {A}}_{t}\left( \begin{array}{c} {\mathbf {x}}_{t} \\ {\mathbf {p}}_{t}\end{array}\right) +\left( \begin{array}{c} {\mathbf {g}}_{t+1} \\ {\mathbf {d}}_{t+1}\end{array}\right) ,\quad t=0,1,2\ldots , \end{aligned}$$

(33)

subject to the restrictions in Eq. (32). In our case, the control vector ${\mathbf {x}}_{t}$ has one variable, the consumption deviation ($\widehat{c}_{t}$), so that $n_{1}=1$. The remaining variables, the capital and wedge deviations, are elements of ${\mathbf {p}}_{t}$.

The next step is to diagonalize ${\mathbf {A}}_{t}$:

$$\begin{aligned} {\mathbf {A}}_{t}={\mathbf {B}}_{t}{\mathbf {J}}_{t}{\mathbf {C}}_{t}, \end{aligned}$$

where: the matrix ${\mathbf {B}}_{t}$ contains the eigenvectors of ${\mathbf {A}}_{t}$; the matrix ${\mathbf {J}}_{t}$ is a diagonal matrix holding the associated eigenvalues; and ${\mathbf {C}}_{t}={\mathbf {B}}_{t}^{-1}$. (In our case, a simple diagonalization always works). Assume that the eigenvalues are sorted by size in descending order, and let $m_{1}$ denote the number of eigenvalues of magnitude greater than 1. In the standard saddle-path case, $m_{1}=n_{1}$. We will hold this assumption throughout our analysis; readers interested in other configurations can consult the references listed above. With saddle-path stability, we can partition ${\mathbf {B}}_{t}$, ${\mathbf {J}}_{t}$, and ${\mathbf {C}}_{t},$ as:

$$\begin{aligned} {\mathbf {B}}_{t} & = \left[ \begin{array}{cc} \underset{\left( n\times n_{1}\right) }{{\mathbf {B}}_{1t}}&\underset{\left( n\times n_{2}\right) }{{\mathbf {B}}_{2t}}\end{array}\right] \equiv \left[ \begin{array}{cc} \underset{\left( n_{1}\times n_{1}\right) }{{\mathbf {B}}_{11t}} & \underset{\left( n_{1}\times n_{2}\right) }{{\mathbf {B}}_{12t}} \\ \underset{\left( n_{2}\times n_{1}\right) }{{\mathbf {B}}_{21t}} & \underset{\left( n_{2}\times n_{2}\right) }{{\mathbf {B}}_{22t}}\end{array}\right] , \\ {\mathbf {J}}_{t} & = \left[ \begin{array}{cc} \underset{\left( n_{1}\times n_{1}\right) }{{\mathbf {J}}_{1t}} & \underset{\left( n_{1}\times n_{2}\right) }{{\mathbf {0}}} \\ \underset{\left( n_{2}\times n_{1}\right) }{{\mathbf {0}}} & \underset{\left( n_{2}\times n_{2}\right) }{{\mathbf {J}}_{2t}}\end{array}\right] , \\ {\mathbf {C}}_{t} & = \left[ \begin{array}{c} \underset{\left( n_{1}\times n\right) }{{\mathbf {C}}_{1t}} \\ \underset{\left( n_{2}\times n\right) }{{\mathbf {C}}_{2t}}\end{array}\right] . \end{aligned}$$

Premultiplying Eq. (33) by ${\mathbf {C}}_{t}$ yields the transformed system

$$\begin{aligned} \widetilde{{\mathbf {w}}}_{t+1}&\equiv \left( \begin{array}{c} \widetilde{{\mathbf {y}}}_{t+1} \\ \widetilde{{\mathbf {q}}}_{t+1}\end{array}\right) =\left[ \begin{array}{cc} {\mathbf {J}}_{1t} & {\mathbf {0}} \\ {\mathbf {0}} & {\mathbf {J}}_{2t}\end{array}\right] \left( \begin{array}{c} {\mathbf {y}}_{t} \\ {\mathbf {q}}_{t}\end{array}\right) +\left( \begin{array}{c} \widetilde{{\mathbf {h}}}_{t+1} \\ \widetilde{{\mathbf {e}}}_{t+1}\end{array}\right) ,\quad t=0,1,2..., \\ {\mathbf {y}}_{t}&={\mathbf {C}}_{1t}{\mathbf {v}}_{t};\quad {\mathbf {q}}_{t}={\mathbf {C}}_{2t}{\mathbf {v}}_{t}, \nonumber \\ \widetilde{{\mathbf {y}}}_{t+1}&={\mathbf {C}}_{1t}{\mathbf {v}}_{t+1};\quad \widetilde{{\mathbf {q}}}_{t}={\mathbf {C}}_{2t}{\mathbf {v}}_{t+1}, \nonumber \\ \widetilde{{\mathbf {h}}}_{t+1}&={\mathbf {C}}_{1t}\left( \begin{array}{c} {\mathbf {g}}_{t+1} \\ {\mathbf {d}}_{t+1}\end{array}\right) ;\quad \widetilde{{\mathbf {e}}}_{t+1}={\mathbf {C}}_{2t}\left( \begin{array}{c} {\mathbf {g}}_{t+1} \\ {\mathbf {d}}_{t+1}\end{array}\right) . \nonumber \end{aligned}$$

(34)

Because the timing of the transformation is important, we use tildes to denote transformed variables with time “mismatches”.

Because the elements of ${\mathbf {J}}_{1t}$ are bigger than 1 in magnitude, the non-explosive solution to the first row of Eq. (34) is to set

$$\begin{aligned} {\mathbf {y}}_{t}=\widetilde{{\mathbf {h}}}_{t+1}=\widetilde{{\mathbf {y}}}_{t+1}={\mathbf {0.}} \end{aligned}$$

It immediately follows that

$$\begin{aligned} {\mathbf {v}}_{t}&= {\mathbf {B}}_{t}{\mathbf {w}}_{t}= {\mathbf {B}}_{2t}{\mathbf {w}}_{t}, \\ {\mathbf {v}}_{t+1}&={\mathbf {B}}_{2t}\widetilde{{\mathbf {w}}}_{t+1}, \\ \left( \begin{array}{c} {\mathbf {f}}_{t+1} \\ {\mathbf {d}}_{t+1}\end{array}\right)&={\mathbf {B}}_{2t}\widetilde{{\mathbf {e}}}_{t+1}=\left[ \begin{array}{c} {\mathbf {B}}_{12t} \\ {\mathbf {B}}_{22t}\end{array}\right] \widetilde{{\mathbf {e}}}_{t+1}. \end{aligned}$$

But because ${\mathbf {d}}_{t+1}$ is given, it must be the case that

$$\begin{aligned} {\mathbf {B}}_{22t}\widetilde{{\mathbf {e}}}_{t+1} & = {\mathbf {d}}_{t+1}, \\ \widetilde{{\mathbf {e}}}_{t+1} & = {\mathbf {B}}_{22t}^{-1}{\mathbf {d}}_{t+1}, \end{aligned}$$

and

$$\begin{aligned} \left( \begin{array}{c} {\mathbf {f}}_{t+1} \\ {\mathbf {d}}_{t+1}\end{array}\right) & = {\mathbf {H}}_{t}{\mathbf {d}}_{t+1}, \\ {\mathbf {H}}_{t} & \equiv \left[ \begin{array}{c} {\mathbf {B}}_{12t}{\mathbf {B}}_{22t}^{-1} \\ {\mathbf {I}}_{n_{2}}\end{array}\right] , \nonumber \end{aligned}$$

(35)

where ${\mathbf {I}}_{n_{2}}$ is an identity matrix of size $n_{2}$.

1.5.2 The effects of time variation

Equation (35) implies that the innovation to the control variable ${\mathbf {x}}_{t}$ is a linear function of the innovations to the state variable ${\mathbf {p}}_{t}$. The same logic, however, applies to the variables ${\mathbf {x}}_{t}$ and ${\mathbf {p}}_{t}$ themselves. The fact that ${\mathbf {p}}_{t}$ is pre-determined at time t, along with the non-explosiveness restriction ${\mathbf {y}}_{t}={\mathbf {0}}$ , implies that

$$\begin{aligned} \left( \begin{array}{c} {\mathbf {x}}_{t} \\ {\mathbf {p}}_{t}\end{array}\right) ={\mathbf {H}}_{t}{\mathbf {p}}_{t}. \end{aligned}$$

(36)

Comparing Eqs. (35) and (36) reveals a timing inconsistency: time-t innovations are “stabilized” using time-$t-1$ coefficients, while the variables themselves are stabilized using time-t coefficients. To see how this plays out, consider the system:

$$\begin{aligned} \left( \begin{array}{c} {\mathbf {x}}_{t+1} \\ {\mathbf {p}}_{t+1}\end{array}\right) ={\mathbf {A}}_{t}\left( \begin{array}{c} {\mathbf {x}}_{t} \\ {\mathbf {p}}_{t}\end{array}\right) +{\mathbf {H}}_{t}{\mathbf {d}}_{t+1}. \end{aligned}$$

(37)

Suppose further that: ${\mathbf {v}}_{0}=$ ${\mathbf {0}}$; ${\mathbf {d}}_{t}={\mathbf {0}}$, $\forall t\ne 1$; and ${\mathbf {d}}_{1}\ne {\mathbf {0}}$. This yields:

$$\begin{aligned} {\mathbf {v}}_{0} & = {\mathbf {0}}, \\ {\mathbf {v}}_{1} & = {\mathbf {H}}_{0}{\mathbf {d}}_{1}, \\ {\mathbf {v}}_{2} & = {\mathbf {A}}_{1}{\mathbf {v}}_{1}={\mathbf {A}}_{1}{\mathbf {H}}_{0}{\mathbf {d}}_{1}, \\ {\mathbf {v}}_{3} & = {\mathbf {A}}_{2}{\mathbf {A}}_{1}{\mathbf {H}}_{0}{\mathbf {d}}_{1} \end{aligned}$$

But it should also be the case that

$$\begin{aligned} {\mathbf {v}}_{2}={\mathbf {A}}_{1}{\mathbf {H}}_{1}{\mathbf {p}}_{1}={\mathbf {A}}_{1}{\mathbf {H}}_{1}{\mathbf {d}}_{1}. \end{aligned}$$

Following Klein (2000), we can show that Eq. (36) generates a bounded solution. In particular,

$$\begin{aligned} {\mathbf {A}}_{t}{\mathbf {H}}_{t} & = \left[ \begin{array}{cc} {\mathbf {B}}_{1t}&{\mathbf {B}}_{2t}\end{array}\right] \left[ \begin{array}{cc} {\mathbf {J}}_{1t} & {\mathbf {0}} \\ {\mathbf {0}} & {\mathbf {J}}_{2t}\end{array}\right] \left[ \begin{array}{c} {\mathbf {C}}_{1t} \\ {\mathbf {C}}_{2t}\end{array}\right] {\mathbf {H}}_{t} \\ & = \left( {\mathbf {B}}_{1t}{\mathbf {J}}_{1t}{\mathbf {C}}_{1t}+{\mathbf {B}}_{2t}{\mathbf {J}}_{2t}{\mathbf {C}}_{2t}\right) {\mathbf {H}}_{t}. \end{aligned}$$

Moreover,

$$\begin{aligned} {\mathbf {H}}_{t}=\left[ \begin{array}{c} {\mathbf {B}}_{12t}{\mathbf {B}}_{22t}^{-1} \\ {\mathbf {I}}_{n_{2}}\end{array}\right] =\left[ \begin{array}{c} {\mathbf {B}}_{12t} \\ {\mathbf {B}}_{22t}\end{array}\right] {\mathbf {B}}_{22t}^{-1}={\mathbf {B}}_{2t}{\mathbf {B}}_{22t}^{-1}. \end{aligned}$$

As a result,

$$\begin{aligned} {\mathbf {A}}_{t}{\mathbf {H}}_{t} & = \left( {\mathbf {B}}_{1t}{\mathbf {J}}_{1t}{\mathbf {C}}_{1t}+{\mathbf {B}}_{2t}{\mathbf {J}}_{2t}{\mathbf {C}}_{2t}\right) {\mathbf {B}}_{2t}{\mathbf {B}}_{22t}^{-1} \\ & = \left( {\mathbf {B}}_{1t}{\mathbf {J}}_{1t}{\mathbf {0}}+{\mathbf {B}}_{2t}{\mathbf {J}}_{2t}{\mathbf {I}}_{n_{2}}\right) {\mathbf {B}}_{22t}^{-1} \\ & = {\mathbf {B}}_{2t}{\mathbf {J}}_{2t}{\mathbf {B}}_{22t}^{-1}, \end{aligned}$$

because, as noted by Klein (2000, p. 1418), ${\mathbf {C}}_{t}{\mathbf {B}}_{t}={\mathbf {I}}$.

Note that

$$\begin{aligned} {\mathbf {A}}_{t}{\mathbf {H}}_{t}={\mathbf {A}}_{t}^{*}{\mathbf {H}}_{t}, \end{aligned}$$

where the “stabilized” transition matrix ${\mathbf {A}}_{t}^{*}$ has been purged of its explosive eigenvalues:

$$\begin{aligned} {\mathbf {A}}_{t}^{*} & = \left[ \begin{array}{cc} {\mathbf {B}}_{1t}&{\mathbf {B}}_{2t}\end{array}\right] \left[ \begin{array}{cc} {\mathbf {0}} & {\mathbf {0}} \\ {\mathbf {0}} & {\mathbf {J}}_{2t}\end{array}\right] \left[ \begin{array}{c} {\mathbf {C}}_{1t} \\ {\mathbf {C}}_{2t}\end{array}\right] \\ & = {\mathbf {B}}_{2t}{\mathbf {J}}_{2t}{\mathbf {C}}_{2t} . \end{aligned}$$

In short, applying Eq. (36) is equivalent to updating Eq. (33) with a non-explosive transition matrix. This result does not hold if we use ${\mathbf {H}}_{t-1}$ from Eq. (35), as ${\mathbf {A}}_{t}$ contains ${\mathbf {C}}_{1t}$, while ${\mathbf {H}}_{t-1}$ contains ${\mathbf {B}}_{2,t-1}$. On the other hand, using Eq. (35) bests captures the transition dynamics in effect at time t. Our solution is this:

• 1. Given ${\mathbf {p}}_{t}$, use Eq. (36) to find ${\mathbf {x}} _{t}$.

• 2. Given $\left( {\mathbf {x}}_{t}^{\prime },{\mathbf {p}}_{t}^{\prime }\right) ^{\prime }$, use the bottom $n_{2}$ rows of Eqs. (33) or (37) to find ${\mathbf {p}}_{t+1}$. Return to step 1.

Using this approach means that ${\mathbf {x}}_{t+1}$ is not entirely consistent with the dynamics implied by Eqs. (33) or (37). In our context, this means that consumption does not perfectly satisfy the linearized Euler equation. The error appears to be less than 1 % of consumption, however, which is small relative to some observed parameter changes.

1.6 Estimated and projected trends

See Figs. 17, 18, 19 and 20.

Fig. 17

Estimated and projected trends: sectoral shares

Fig. 18

Estimated and projected trends: sector-specific total factor productivity

Fig. 19

Estimated and projected trends: capital and labor market distortions

Fig. 20

Estimated and projected trends: fiscal policies and depreciation rate